Equip this category with the coverage where a family of morphisms is covering precisely if it is of the form {Ui×D→(fi,IdD)U×D}\{U_i \times D \stackrel{(f_i, Id_D)}{\to} U \times D\} for {fi:Ui→U}\{f_i : U_i \to U\} a covering in CartSpsmooth{}_{smooth} (a good open cover).

Structures

Since by the above discussionFormalSmooth∞GrpdFormalSmooth\infty Grpd is strongly ∞\infty-connected relative to Smooth∞Grpd all of these structures that depend only on ∞\infty-connectedness (and not on locality) acquire a relative version.

Proposition

For X∈SmoothAlgop→FormalSmooth∞GrpdX \in SmoothAlg^{op} \to FormalSmooth \infty Grpd a smooth locus, we have that Πinf(X)\mathbf{\Pi}_{inf}(X) is the corresponding de Rham space, the object in which all infinitesimal neighbour points in XX are equivalent, characterized by

Πinf(X):U×D↦X(U).
\mathbf{\Pi}_{inf}(X) : U \times D \mapsto X(U)
\,.

Proof

By the (Red⊣Πinf)(\mathbf{Red} \dashv \mathbf{\Pi}_{inf})-adjunction relation

Proof

First a remark on the sites. By the above propositionFormalSmooth∞GrpdFormalSmooth\infty Grpd is equivalent to the hypercomplete (∞,1)-topos over formal smooth manifolds. This is presented by the left Bousfield localization of [FSmoothMfdop,sSet]proj,loc[FSmoothMfd^{op}, sSet]_{proj,loc} at the ∞-connected morphisms. But a fibrant object in [FSmoothMfdop,sSet]proj,loc[FSmoothMfd^{op}, sSet]_{proj,loc} that is also n-truncated for n∈ℕn \in \mathbb{N} is also fibrant in the hyperlocalization (only for the untruncated object there is an additional condition). Therefore the right Quillen functor claimed above indeed lands in truncated objects in FormalSmoothinftyGrpdFormalSmooth \inftyGrpd.

Corollary

Proof

This follows from applying the above result to the fiber sequence induced by the sequence ℤ→ℝ→ℝ/ℤ=U(1)\mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} = U(1).

Note

This means that the intrinsic cohomology of compactLie groups in Smooth∞Grpd and FormalSmooth∞GrpdFormalSmooth\infty Grpd coincides for these coefficients with the Segal-Blanc-Brylinski refined Lie group cohomology (Brylinski).

Observe that 𝒪(𝔞)•\mathcal{O}(\mathfrak{a})_\bullet is cofibrant in the Reedy model structure [Δop,(SmoothAlgprojΔ)op]Reedy[\Delta^{\mathrm{op}}, (\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}]_{\mathrm{Reedy}} relative to the opposite of the projective model structure on cosimplicial algebras:the map from the latching object in degree nn in SmoothAlgΔ)op\mathrm{SmoothAlg}^\Delta)^{\mathrm{op}} is dually in SmoothAlg↪SmoothAlgΔ\mathrm{SmoothAlg} \hookrightarrow \mathrm{SmoothAlg}^\Delta the projection

hence is a surjection, hence a fibration in SmoothAlgprojΔ\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}} and therefore indeed a cofibration in (SmoothAlgprojΔ)op(\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}.

Therefore using the Quillen bifunctor property of the coend over the tensoring in reverse to this lemma, the above is equivalent to

It follows that a degree-nnℝ\mathbb{R}-cocycle on 𝔞\mathfrak{a} is presented by a morphism

μ:𝔞→bnℝ,
\mu : \mathfrak{a} \to b^n \mathbb{R}
\,,

where on the right we have the L∞L_\infty-algebroid whose CE\mathrm{CE}-algebra is concentrated in degree nn. Notice that if 𝔞=b𝔤\mathfrak{a} = b \mathfrak{g} is the delooping of an L∞L_\infty- algebra 𝔤\mathfrak{g} this is equivalently a morphism of L∞L_\infty-algebras

μ:𝔤→bn−1ℝ.
\mu : \mathfrak{g} \to b^{n-1} \mathbb{R}
\,.

de Rham theorem

under construction

We consider the de Rham theorem in FormalSmooth∞GrpdFormalSmooth \infty Grpd, with respect to the infinitesimal de Rham cohomology

Therefore a morphism Πinf(U)→ℝ\mathbf{\Pi}_{inf}(U) \to \mathbb{R} is equivalently a morphism ϕ:U→ℝ\phi : U \to \mathbb{R} such that for all K×D→UK \times D \to U that coincide on KK we have that all the composites

are equals. If UU is the point, then ϕ\phi is necessarily constant. If UU is not the point, there is one nontrivial tangent vector vv in UU. The composite produces the corresponding tangent vector ϕ*(v)\phi_*(v) in ℝ\mathbb{R}. But all these tangent vectors must be equal. Hence ϕ\phi must be constant.

Proposition

Then j!X•j_! X_\bullet in FormalSmooth∞GrpdFormalSmooth \infty Grpd is presented by the same simplicial manifold.

Proof

First consider an ordinary smooth paracompact manifold XX. It admits a good open cover{Ui→X}\{U_i \to X\} and the corresponding Cech nerveC({Ui})in[CartSpsmoothop,sSet]projC(\{U_i\}) in [CartSp_{smooth}^{op}, sSet]_{proj} is a cofibrant resolution of XX. Therefore the ∞\infty-functor j!j_! is computed on XX by evaluating the corresponding simplicial functor (of which it is the derived functor) on C({Ui})C(\{U_i\}).

Here we used that, by assumption on a good open cover, all the Ui0,⋯,inU_{i_0, \cdots, i_n} are Cartesian spaces, and that j!j_! coincides on representables with the inclusion CartSpsmooth↪CartSpformalsmoothCartSp_{smooth} \hookrightarrow CartSp_{formalsmooth}.

Let now X•X_\bullet be a general simplicial manifold. Assume that in each degree there is a good open cover{Up,i→Xp}\{U_{p,i} \to X_p\} such that these fit into a simplicial system giving a bisimplicial Cech nerve such that there is a morphism of bisimplicial presheaves

C(𝒰)•,•→X•
C(\mathcal{U})_{\bullet, \bullet} \to X_{\bullet}

with X•X_\bullet regarded as simplicially constant in one direction. Each C(𝒰)p,•→XpC(\mathcal{U})_{p, \bullet} \to X_p is a cofibrant resolution.

is a cofibrant resolution of XX, where Δ\mathbf{\Delta} is the fat simplex. From this the proposition follows by again applying the left Quillen functor j!j_! degreewise and pulling it through all the colimits.

This remaining claim follows from the same argument as used above in prop. 9.

Remark

Since i*i^* preserves ∞\infty-limits, this is the case in particular if the diagram is an ∞\infty-pullback already in FormalSmooth∞GrodFormalSmooth\infty Grod. In this form, restricted to 0-truncated objects, hence to the Cahiers topos, this characterization of formally étale morphisms appears axiomatized around p. 82 of (Kock 81, p. 82).

is a pullback in Sh(CartSpformalsmooth)Sh(CartSp_{formalsmooth}) precisely if ff is a local diffeomorphism. This is a pullback precisely if for all U×D∈CartSpsmooth⋉InfPoint≃CartSpformalsmoothU \times D \in CartSp_{smooth} \ltimes InfPoint \simeq CartSp_{formalsmooth} the diagram of sets of plots

is a pullback. Using, by the discussion at ∞-cohesive site, that i!i_! preserves colimits and restricts to ii on representables, and using that i*(U×D)=Ui^* (U \times D ) = U, this is equivalently the diagram

where the vertical morphisms are given by restriction along the inclusion (idU,*):U→U×D(id_U, *) : U \to U \times D.

For one direction of the claim it is sufficient to consider this situation for U=*U = * the point and DD the first order infinitesimal interval. Then Hom(*,X)Hom(*,X) is the underlying set of points of the manifold XX and Hom(D,X)Hom(D,X) is the set of tangent vectors, the set of points of the tangent bundleTXT X. The pullback

Hom(*,X)×Hom(*,Y)Hom(D,Y)
Hom(*,X) \times_{Hom(*,Y)} Hom(D,Y)

is therefore the set of pairs consisting of a point x∈Xx \in X and a tangent vector v∈Tf(x)Yv \in T_{f(x)} Y. This set is in fiberwise bijection with Hom(D,X)=TXHom(D, X) = T X precisely if for each x∈Xx \in X there is a bijection TxX≃Tf(x)YT_x X \simeq T_{f(x)}Y , hence precisely if ff is a local diffeomorphism. Therefore ff being a local diffeo is necessary for ff being formally étale with respect to the given notion of infinitesimal cohesion.

To see that this is also sufficient notice that this is evident for the case that ff is in fact a monomorphism, and that since smooth functions are characterized locally, we can reduce the general case to that case.

Proposition

Proof

Let 𝒢0→𝒢\mathcal{G}_0 \to \mathcal{G} be the inclusion of the smooth manifold of objects. This is an effective epimorphism. It remains to show that this is formally étale with respect to the given cohesive neighbourhood.

on those objects whose space of global sections is contractible and which are infinitesimally cohesive (for a somewhat different notion of “infinitesimal cohesion”, beware the terminology). Consider then the ∞\infty-functor

The cohomology localization of SynthDiff∞GrpdSynthDiff\infty Grpd and the infinitesimal singular simplicial complex as a presentation for infinitesimal paths objects in SynthDiff∞GrpdSynthDiff\infty Grpd is discussed in